Q: If $$\delta(x, y) is $$the density at point $$(x, y)$$, what is the definition of density in terms of mass and area?

$$\delta(x, y) = \lim_{\Delta A \to 0} \frac{\Delta m}{\Delta A}$$

Question 1

A uniform thin plate in the shape of a region bounded by $$y = 2x$$, $$x = 1$$, and the $$x-$$axis has a constant density $$\delta = 3\ \mathrm{g/cm}^2. $$What is the total mass $$M of $$the plate?

Accepted Answer

$$M = \$$int_0$$^1 3 \cdot 2x\ dx = 3$$

Question 2

Given two masses $$m_1$$ and $$m_2$$ located at $$x_1$$ and $$x_2 on $$the $$x-$$axis, what is the formula for the $$x-$$coordinate of the center of mass?

Accepted Answer

$$\overline{x} = \frac{x_1$ m_1$ + x_2$ m_2$}{m_1$ + m_2$}$$

Question 3

A seesaw is balanced with a child of weight $$W_1$$ sitting at distance $$d_1$$ from the fulcrum and another child of weight $$W_2 at $$distance $$d_2 on $$the other side. What is the condition for balance?

Accepted Answer

$$d_1$$ $$W_1$$ = $$d_2$$ $$W_2$$

Question 4

For a region bounded by $$y = f(x)$$, $$y = g(x)$$, $$x = a$$, and $$x = b$$, with constant density $$\delta$$, what is the formula for the $$x-$$coordinate of the center of mass?

Accepted Answer

$$\overline{x} = \frac{1}{M} \int_a^b \delta x [f(x) - g(x)]\ dx$$

Question 5

If the density $$\delta(x, y) of a $$plate is not constant, how is the mass $$dm of a $$small area $$dA$$ expressed?

Accepted Answer

$$dm = \delta(x, y)\ dA$$

Question 6

Which of the following statements is true about the center of mass $$\overline{x}$$ for a plate with uniform density?

Accepted Answer

$$\overline{x}$$ depends only on the shape of the plate, not the material.

Question 7

Given $$\$$int_1$$^\infty \frac{dx}{x^p}$$, for which values of $$p$$ does the improper integral converge?

Accepted Answer

$$p \u003e 1$$

Question 8

Evaluate $$\int_{-\infty}^{\infty} \frac{dx}{1 + x^2$}.$$

Accepted Answer

$$\pi$$

Question 9

Which of the following best describes an improper integral of Type II?

Accepted Answer

An integral where the integrand becomes infinite at a point within the interval of integration.

Question 10

Suppose a plate is cut into several pieces, each with its own center of mass. If the connections between the pieces are rigid and weightless, what can we say about the system?

Accepted Answer

The system's center of mass can be found by treating each piece as a point mass at its center of mass.

Question 11

Given $$\$$int_1$$^b \frac{\ln x}{x^2$} dx = -\frac{\ln b}{b} - \frac{1}{b} + 1$$, what is $$\$$int_1$$^{\infty} \frac{\ln x}{x^2$} dx$$?

Accepted Answer

$$1$$

Question 12

A region is bounded above by $$y = f(x)$$ and below by $$y = g(x)$$ from $$x = a to$$ $$x = b. $$What is the formula for the total mass $$M if $$the density is $$\delta$$?

Accepted Answer

$$M = \int_a^b \delta [f(x) - g(x)] dx$$

Question 13

If $$\delta(x, y) is $$the density at point $$(x, y)$$, what is the definition of density in terms of mass and area?

Accepted Answer

$$\delta(x, y) = \lim_{\Delta A 	o 0} \frac{\Delta m}{\Delta A}$$

Question 14

Evaluate the improper integral $$\$$int_0$$^1 \frac{dx}{(x-1$$)^{2/3}$$}.$$

Accepted Answer

$$3$$

Question 15

For a region bounded by $$y = f(x)$$ and $$y = g(x)$$, what is the formula for the $$y-$$coordinate of the center of mass?

Accepted Answer

$$\overline{y} = \frac{1}{M} \int_a^b \frac{\delta}{2} [f^2$(x) - g^2$(x)] dx$$

Moments, Centers of Mass, and Improper Integrals

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